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Election questions

Gaffes, murder plots, and hockey mums — the US election circus is in full swing. Personality politics is partly a consequence of the first-past-the-post system — but what are the alternatives? Can any system guarantee to represent the will of the people? Do opinion polls manage to capture its voice? The following selection of Plus articles provides mathematical answers, insights and
paradoxes.

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History of maths writing competition

The History of Mathematics Special Interest Group of the Mathematical Association of America (HOMSIGMAA) has just launched a writing competition for undergraduates. The MAA invites students to write on any topic in mathematical history they choose, and winners will receive a year-long MAA student membership, as well as books from the MAA bookstore. Closing date is the 31st of March 2009. For
more information, visit the HOMSIGMAA website.

And if you're not a student, or want to write about something other than history, or simply want to be published in Plus, then don't forget our very own Plus New Writers Award 2009. Besides the fame and glory of appearing in Plus, you could win an iPod and selected books. Closing date 31st of March 2009!

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Plus is proud to bring you a brand new regular feature: the Plus sports page. This will be regularly updated with stories exploring the maths in sports, from analysing scoring statistics to planning the perfect strategy.

We start off by looking at — what else? — football. Being the manager of a Premier League football club may seem like one of the most glamorous jobs in the world — with the fame comes fortune and the opportunity to travel (well, to Hull, Wigan and Portsmouth anyway). However, as far as job security goes, football managers live on the edge. Their terms can be terminated almost on a whim by
their club's owner, and they live and die by their team's results.

It would seem that there is no way to predict how long their tenures will be. However, a collection of researchers from the UK, Singapore and the US have found that there may be a strong mathematical trend underlying how long football managers stay in their jobs.

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Really interesting results and an enjoyable article. I think an interesting sideline could be the relationship between the chairmans length of tenure and the managers. Clubs also have a reputation; take the Newcastle United fiasco recently. Did the researchers find a link between the prior success of the club under the previous manager, with the length of tenure of subsequent managers? i.e.,
when Alex Ferguson leaves Manchester United, will his successor last as long?

Mathematical moments - How to price derivatives

The 1997 Nobel Prize in Economics

It's hard to believe these days, but bankers do occasionally base their decisions on rational and robust bits of mathematics. A central player in this context is the Black-Scholes equation, which won its discoverers the Nobel Prize in Economics almost exactly 11 years ago, on the 14th of October 1997. The Black-Scholes equation is a partial differential equation which tells you how to price
options. An option is a way of reducing risk: it's a forward contract giving you the opportunity (but not necessarily the obligation) to buy a certain commodity or stock, for example fuel, at some specified time in the future for an agreed strike price. An option is an example of a so-called derivative, because it derives its value from the underlying variable (the price of the fuel).

Derivatives and options, as we all now know, can be traded in their own right, and have the potential to make people very rich, but also to break the bank (as happened literally with the Barings Bank in 1995). One important question if you're in the business of trading derivatives, is how much they should cost. Since options are all about the future, it may seem as if pricing them should
involve a certain amount clairvoyancy, but in fact the Black-Scholes equation does it all for you (for a type of option called a European call). All you need to know to use the equation are the current price of the stock and strike price of the option, the expiry time of the option, the interest rate and an estimate of the versatility of the stock.

In their original 1973 paper Black and Scholes built their model on the assumption that the stock price was a function of a Brownian motion. A Brownian motion is a random motion, as you would expect to see when you let a particle float in a liquid or gas. Mathematically, you can describe it as a stochastic process: at any given time, the stock price (or the particle) moves up or down (or, with
a particle, in any of the available directions) with equal probability. Using this model for stock price behaviour, Black and Scholes found that the problem of pricing an option becomes equivalent to that of solving the heat equation from physics.

The original Black-Scholes model made some simplifying assumptions, for example that shares don't pay dividends, but it is possible to work around many of these and obtain more sophisticated and comprehensive models. These have been at the heart of financial mathematics ever since. Unfortunately, Fischer Black died two years before the award of the Nobel Prize, so Myron S. Scholes and Robert
C. Merton (who's name isn't attached to the equation) ended up sharing it.

If all of this makes you wonder who should really be blamed for the current financial crisis, read the Plus article Is maths to blame?

For more in-depth information on derivatives, read the Plus article Rogue trading?

The Internet is huge. With mobile phones and other devices now being hooked up as well as computers, it will soon comprise many billions of endpoints. In sheer size and complexity the Internet is not far off the human brain with its hundred billion neurons linked up by around ten thousand trillion individual connections. If you're finding it hard even to read these huge numbers off the page,
how can anyone be expected to cope with complexity on this vast scale? While mathematical tools that deal with complexity have experienced rapid development it recent years, there still isn't an overarching science of complexity, a mathematical toolbox serving everyone who's dealing with it in whatever shape or form.

This month a group of experts from a range of different areas got together in Cambridge to try and remedy the situation. It was the first ever meeting of the Cambridge Complex Systems Consortium. Delegates included a climate scientist, a former head of MI6, a neuroscientist, a sociologists, and a mathematician. Plus went to see them, to learn about their struggle with
complexity.